Celebratio Mathematica

G. Kuperberg and K. Kuperberg: “Generalized counterexamples to the Seifert conjecture,” Ann. Math. (2) 143 : 3 (May 1996), pp. 547–576. A corrected version was published in Ann. Math. 144:2 (1996). MR 1394969 article

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere \( S^3 \) has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on \( S^3 \) with no singular points has a periodic trajectory.

G. Kuperberg and K. Kuperberg: “Generalized counterexamples to the Seifert conjecture,” Ann. Math. (2) 144 : 2 (September 1996), pp. 239–268. This is a corrected version of the article originally published in Ann. Math. 143:3 (1996). MR 1418899 Zbl 0856.57026 ArXiv math/9802040 article

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere \( S^3 \) has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on \( S^3 \) with no singular points has a periodic trajectory.

K. Kuperberg: “Counterexamples to the Seifert conjecture,” pp. 831–840 in Proceedings of the International Congress of Mathematicians, volume 2: Invited lectures (Berlin, 18–27 August 1998), published as Doc. Math. Extra Volume. Issue edited by G. Fischer and U. Rehmann. Deutsche Mathematiker-Vereinigung (Berlin), 1998. Dedicated to the author’s son Greg. MR 1648130 Zbl 0924.58086 incollection

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G. Kuperberg, K. Kuperberg, and W. Kuperberg: “Lattice packings with gap defects are not completely saturated,” Beitr. Algebra Geom. 45 : 1 (2004), pp. 267–273. MR 2070648 Zbl 1054.52010 ArXiv math/0303366 article

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